The following question below asks to find a vector $w$ and then a vector $u_{3}$ given a set of vectors and the information provided below. The vector calculations I can manage, but I seem to be getting tripped up on the orthonormal condition that the question asks for. Any advice or tips on approaching this problem would be highly appreciated.
Given the vectors;
$$ u_{1}=\frac{1}{\sqrt{3}} \begin{bmatrix}
1 \\
-1 \\
1
\end{bmatrix},\ u_{2}=\frac{1}{\sqrt{6}} \begin{bmatrix}
1 \\
2 \\
1
\end{bmatrix},\
v= \begin{bmatrix}
1 \\
2 \\
3
\end{bmatrix}$$
Calculate the vector
$$ w = v − \langle v,u_{1} \rangle u_{1} – \langle v,u_{2} \rangle u_{} $$
and hence find a vector $u_{3} ∈ R^{3}$, such that $ \{ u1, u2, u3 \} $ is an orthonormal set of vectors.
Best Answer
If you have computed $w$, then you're almost done: the set$$\left\{u_1,u_2,\frac w{\lVert w\rVert}\right\}$$will be an orthonormal set of vectors.